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Journal of Spectral Theory

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Volume 4, Issue 2, 2014, pp. 309–347
DOI: 10.4171/JST/71

Published online: 2014-07-13

Sharp spectral bounds on starlike domains

Richard S. Laugesen[1] and Bartłomiej A. Siudeja[2]

(1) University of Illinois at Urbana-Champaign, USA
(2) University of Oregon, Eugene, USA

We prove sharp bounds on eigenvalues of the Laplacian that complement the Faber–Krahn and Luttinger inequalities. In particular, we prove that the ball maximizes the first eigenvalue and minimizes the spectral zeta function and heat trace. The normalization on the domain incorporates volume and a computable geometric factor that measures the deviation of the domain from roundness, in terms of moment of inertia and a support functional introduced by Pólya and Szegó.

Additional functionals handled by our method include finite sums and products of eigenvalues. The results hold on convex and starlike domains, and for Dirichlet, Neumann or Robin boundary conditions.

Keywords: Isoperimetric, membrane, convex, spectral zeta, heat trace, partition function, sloshing

Laugesen Richard, Siudeja Bartłomiej: Sharp spectral bounds on starlike domains. J. Spectr. Theory 4 (2014), 309-347. doi: 10.4171/JST/71